Let S and G be finite groups of coprime order such that S acts on G. If S is solvable, Glauberman [11] proves the existence of a bijection between the S-fixed irreducible representations of G and the irreducible representations of Gs. In the case of G solvable, Isaacs [13] uses a totally different method to prove the existence of a bijection between the same two sets of representations. Assuming the existence of the Glauberman correspondence, Boltje [5] uses the method of Explicit Brauer Induction (EBI) to give an explicit version of this correspondence for the case in which S is a p-group. After presenting the above results, we outline a strategy for investigating these correspondences using Explicit Brauer Induction, and we use these ideas to give a new proof for the theorems of Glauberman and Boltje. We move on to suggest some ideas of how this work may extend to Isaacs' correspondence. We also mention a link to Shintani's correspondence [25]. In the final chapter, we look at cryptography, and mention a potential application of some of our techniques (Adams Operations) in this field.